Efficient high order method for differential equations in unbounded domains using generalized coordinate transformation

Many physical phenomena are localized in space, where physical quantities of interest extend to large distances (effectively to infinity) where they vanish smoothly. Thus, its modeling necessitates the imposition of vanishing boundary conditions (BCs) to the formulated differential equations (DEs). Numerical solutions for such DEs can be complicated as they involve operations on physical quantities expanding infinitely in space. Currently, many approaches have been implemented in this context, where they mostly use techniques like domain truncation, using functions intrinsic to unbounded domains, and domain transformation. However, since the actual solutions have a decaying nature, solving such problems with high accuracy and efficiency while taking into consideration a wide range of decaying rates and oscillation frequencies is a major challenge. In this work, we present an efficient high-order method, based on Galerkin method, to solve DEs in unbounded domains using properly designed domain transformations where the physical unbounded domain (−∞,∞) is mapped into a computational bounded domain (0,1). The mapped sine series is then used to solve the differential equation in the computational domain (0,1) where the vanishing quantities at infinities are mapped to 0 (corresponding to physical −∞) and 1 (corresponding to physical ∞). The designed transformations maintain the orthogonality in both physical and computational spaces. Moreover, using sine series basis allows – in most cases – analytical integrations. For verification, the proposed method is applied to solve 1) Helmholtz equation in one-dimension, where the exact solutions for various cases (decaying exponential or algebraic function) are known, and 2) three-dimensional Schrödinger wave equation with harmonic oscillator. For the Helmholtz problem, the error in the approximate solution was less than 10−40 using 100 basis functions, while for the second problem, the error is less than 10−25 using the same number of bases. All results and the analyses demonstrate how the properly designed mapped basis sets can be used to develop efficient high-order methods to solve DEs with vanishing boundary conditions and with wide range of decaying rates and oscillation frequencies simultaneously.